VW = KW (UI , U W ) (FW - ∇U W )
water are of concern. At this scale we refer to the
(B21)
porous medium behaving as a continuum display-
in which the reduced volumetric flux of water
ing uniform material properties, and we are inter-
VW is, by balancing constants,
ested in the flow into and then out of a volume
of material. This suggests that a different length
ζη
scale is required to describe the gradient, since λ
VW ≡
vW .
(B22)
λγ IA
is applicable only at the pore scale. The additional
macroscopic length scale is ζ. Using this new
The continuity equation for pore constituents
length scale to reduce the gradient operator re-
is
sults in the reduced body force:
∇ (vW + YIWvI )
ζλ
FW ≡
fW .
(B18)
γ AW
[ W (uW , uI ) + YIW I (uW , uI ) ]
=-
(B23)
The proportionality factor in Darcy's equation
t
is the source of much confusion. In eq B17 it is
in which vI (m3/m2s) is the volumetric flux of
ice and t is time. The operator ∇ is the diver-
entists tend to use hydraulic conductivity K (m/
gence, which simply scales by multiplying by ζ.
s), and geologists and petroleum engineers use
intrinsic permeability k (m2). These three param-
The volumetric flux of ice must scale like the
eters are related by
volumetric flux of water to maintain dimensional
equality on the left side of eq B23. Since the volu-
K
k
kW =
=
(B19)
metric water and ice contents are already dimen-
ρg η
sionless, the reduced time T is
in which ρ (kg/m3) is the density of water, g is
λγ IA
T≡
t.
(B24)
the acceleration due to gravity (or other body
ζ2η
force) and η (N/m2s) is the viscosity of water.
They are all functions of water and ice content,
The conservation of thermal energy is
which are functions of the water and ice pres-
[ CH (uW , uI ) θ
∇ (vH + hIWvW ) = -
sures.
t
The process of scaling the capillary conduc-
tivity consists of first realizing that it is already
]
θ + hIWW (uW , uI )
(B25)
inversely proportional to viscosity (i.e., eq B19),
so it must therefore be scaled by multiplying by
in which vH (W/m2) is the macroscopic flux of sen-
the viscosity. It then becomes the intrinsic perme-
sible heat and CH(uW, ur) (J/m3 K) is the volumet-
ability, which is in dimensions of length squared.
To complete the scaling we must decide on the
ric heat capacity excluding heat resulting from
correct length scale to use. Again relying on the
phase change. The volumetric heat capacity is as-
continuum approximation, the conductivity
sumed to be a material property that is constant
should not depend on the size of the sample once
in the continuum assumption for a given state of
it attains the minimum size required for the con-
ice and water contents. Starting with the left side
tinuum approximation. This requires the use of
of eq B25 the reduced macroscopic flux of sen-
the microscopic length scale λ to scale the con-
sible heat VH is obtained
ductivity. Miller and Miller (1955) deduced the
ζη
same length scale choice by considering the
VH ≡
vH .
(B26)
γ IA
2
NavierStokes equation applied to flow through
the pore. The reduced capillary conductivity is
The reduced volumetric heat capacity CH must
then
therefore be
η
KW ≡
kW .
(B20)
λθo
CH (U W , UI ) ≡
γ IA H ( W I )
λ2
C u ,u .
(B27)
Substituting eq B20, B19, B6 and the scaling fac-
tor ζ to reduce the gradient operator into eq B17
The macroscopic flux of sensible heat is given
gives the scaled form of Darcy's equation:
by Fourier 's law:
16
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